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On the Capacity of Decode-and-Forward Relaying over Rician Fading Channels

Manav R. Bhatnagar

arXiv:1407.1112v1cs.ITmath.IT

TL;DR

Prior adaptive-capacity studies cited here focus on AF relaying, while the cited DF modulation study concerns Rayleigh fading. This paper derives power-series distributions for the weakest link and uses them to obtain exact DF ergodic capacities over dissimilar Rician channels. Optimal simultaneous power and rate adaptation gains over constant-power rate adaptation at low to moderate SNR, while channel inversion is generally poorer.

  • Problem

    The cited adaptive-capacity work on Rician fading concerns AF, while the cited DF adaptive-modulation study concerns Rayleigh fading.

    Source: §I · p. 1

  • Method

    The paper derives power-series PDF and CDF expressions for the minimum of two non-central Chi-square variables and uses them to obtain exact average capacity for adaptive DF over dissimilar Rician channels.

    Source: §I · p. 1

  • Results

    Optimal simultaneous power and rate adaptation gains capacity over constant-power optimal rate adaptation at low to moderate SNR, while channel inversion is poorer than the other two schemes in general.

    Source: §IV · p. 4

  • Takeaways & Limitations

    At high SNR, optimal rate adaptation with constant power can be the lower-complexity choice because simultaneous power and rate adaptation provides almost no capacity improvement.

    Source: §IV · p. 4

Abstract

from arXiv · show

In this letter, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the minimum of two non-central Chi-square random variables with two degrees of freedom in terms of power series. With the help of the derived PDF and CDF, we obtain the exact ergodic capacity of the following adaptive protocols in a decode-and-forward (DF) cooperative system over dissimilar Rician fading channels: (i) constant power with optimal rate adaptation; (ii) optimal simultaneous power and rate adaptation; (iii) channel inversion with fixed rate. By using the analytical expressions of the capacity, it is observed that the optimal power and rate adaptation provides better capacity than the optimal rate adaptation with constant power from low to moderate signal-to-noise ratio values over dissimilar Rician fading channels. Despite low complexity, the channel inversion based adaptive transmission is shown to suffer from significant loss in capacity as compared to the other adaptive transmission based techniques over DF Rician channels.

I. INTRODUCTION

The paper addresses adaptive capacity analysis for decode-and-forward (DF) relaying over dissimilar Rician channels, where prior adaptive-capacity studies cited here focus on amplify-and-forward (AF). It derives power-series distributions for the weakest link and uses them to obtain exact DF capacity expressions and compare adaptive schemes.

I. INTRODUCTION

  • The cited adaptive-capacity analyses focus on AF, while the cited DF modulation study concerns Rayleigh fading, motivating exact adaptive-capacity analysis for DF over dissimilar Rician channels.The paper notes that DF is useful because it requires digital processing at the relay.Source: §I · p. 1
  • The paper derives the PDF and CDF of the minimum of two non-central Chi-square random variables with two degrees of freedom as converging power series.The link SNRs are independent and non-identically distributed; the cited normal-variable construction also relates to the two-degree-of-freedom distribution.Source: §I · p. 1, §II · p. 1
  • The derived distributions yield exact average capacity for a two-hop DF system with adaptive modulation over dissimilar Rician channels.The paper uses the expressions to compare adaptive transmission techniques.Source: §I · p. 1
  • The CDF and PDF power series converge, and the CDF expression converges to the simulated CDF for a finite number of summation terms.Figure 1 varies Kx, Ky, and the number of summation terms with average SNRs equal to 5.Source: §II · p. 2, Figure 1 · p. 2

III. SYSTEM MODEL AND CAPACITY ANALYSIS UNDER ADAPTIVE TRANSMISSION

The system is a single-relay, half-duplex DF link with no direct source-destination path and two orthogonal channels. Its capacity-equivalent SNR is determined by the weaker of the two hops.

III. SYSTEM MODEL AND CAPACITY ANALYSIS UNDER ADAPTIVE TRANSMISSION

  • The model assumes a single half-duplex relay and no direct link between source and destination.Source: §III · p. 2
  • The source-to-relay and relay-to-destination links are independent, non-identically distributed Rician channels with respective LOS components Kx and Ky and average SNRs.Each instantaneous link SNR is modeled as a two-degree-of-freedom non-central Chi-square random variable.Source: §III · p. 2
  • Because DF capacity is a monotonous function of SNR, the system is capacity-equivalent to the weakest hop, whose SNR is the minimum of the two hop SNRs.The relay decodes and forwards between the source and destination over two orthogonal channels.Source: §III · p. 2

A. Optimal Rate Adaptation with Constant Transmit Power

The section gives the channel capacity for optimal rate adaptation with constant transmit power. It obtains the expression by substituting the SNR density and applying a lemma.

A. Optimal Rate Adaptation with Constant Transmit Power

B. Optimal Simultaneous Power and Rate Adaptation

The section derives capacity for optimal simultaneous power and rate adaptation under an average transmit-power constraint. It uses a cutoff SNR below which transmission is suspended.

B. Optimal Simultaneous Power and Rate Adaptation

C. Channel Inversion with Fixed Rate

The section presents truncated channel inversion with fixed rate as a low-complexity adaptive transmission technique. It adjusts power to maintain a constant destination SNR and operates only above a cutoff.

C. Channel Inversion with Fixed Rate

IV. NUMERICAL RESULTS AND DISCUSSION

The numerical results show that joint power and rate adaptation gains capacity over rate adaptation with constant power at low to moderate SNR, but offers almost no gain at high SNR. Channel inversion performs more poorly than the other schemes in general.

IV. NUMERICAL RESULTS AND DISCUSSION

  • Figure 2 simulations closely match the analytical capacity values for adaptive rate with constant power.Source: §IV · p. 3
  • Joint power-and-rate adaptation gains capacity at low to moderate SNR, but provides almost no gain over rate adaptation with constant power at high SNR.Figure 3 compares the adaptive schemes over different Rician K values.Source: §IV · p. 4
  • Channel inversion has poorer optimized capacity than the other two schemes in general under the constraint β0 ≤ 1.The stated constraint limits the DF system's maximum outage probability.Source: §IV · p. 4
  • Average SNR ¯γ is shown on the x-axis of Figures 2 and 3.Source: §IV · p. 4

APPENDIX A PROOF OF LEMMA 1 AND LEMMA 2

The appendix derives the CDF and PDF of γz, using series representations and algebraic identities to obtain the stated expressions.

APPENDIX A PROOF OF LEMMA 1 AND LEMMA 2

  • The CDF of γz is obtained after algebraic manipulation.Source: Appendix A · p. 4
  • The PDFs of γx and γy are written using the series representation of the modified Bessel function.The derivation cites [15, Eq. (2.3.32)].Source: Appendix A · p. 4
  • Differentiating the CDF expression in (3) with respect to γ gives the PDF of γz.The derivation uses (20) and [13, Eq. (0.316)] in (18) to obtain (3).Source: Appendix A · p. 4
  • The derivation further simplifies the expression using a relation for the upper incomplete gamma function when r is a positive integer.The relation is identified as (4).Source: Appendix A · p. 4

APPENDIX B PROOF OF LEMMA 3

The appendix proves Lemma 3 by transforming an expression and applying cited relations to recover the right-hand side of (6).

APPENDIX B PROOF OF LEMMA 3

  • The proof begins by denoting the left-hand side of (6).Source: Appendix B · p. 4
  • Substituting 1 + w = t into the right-hand side of (22) yields a transformed expression.Source: Appendix B · p. 4
  • The proof applies [13, Eq. (1.111)] and [13, Eq. (4.358.1)] along with the relation defining f(q) = q + 1.Source: Appendix B · p. 4
  • Using (26) in (25) and algebraic manipulation gives the right-hand side of (6).Source: Appendix B · p. 4

APPENDIX C PROOF OF LEMMA 4

The appendix proves Lemma 4 by substituting t = w/η and applying cited relations.

APPENDIX C PROOF OF LEMMA 4

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